Papers

A deformation of Heisenberg algebra induces among other consequences a loss of Hermiticity of some operators that generate this algebra. Therefore, these operators are not Hermitian, nor is the Hamiltonian operator built from them. In the present paper, we propose a position deformation of Heisenberg algebra with both maximal length and minimal momentum uncertainties. By using a pseudo-similarity transformation to the non-Hermitian operators, we prove their Hermiticity with a suitable positive-definite pseudo-metric operator. We then construct Hilbert space representations associated with these pseudo-Hermitian operators. Finally, we study the eigenvalue problem of a free particle in this deformed space and we show that this deformation curved the quantum levels allowing particles to jump from one state to another with low energy transitions.

We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors. This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions. We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.

We consider the recent proposal of steady state coherences in a single qubit in the case of a composite system-bath interaction. Based on a field theoretical approach we reanalyse the issue within a Redfield description. We find that the Redfield approach in accordance with a recent proposal yields steady state coherences but also violates the properties of a quantum map yielding negative populations. The issue is resolved by applying the Lindblad equation which is in accordance with a proper quantum map. The Lindblad equation, however, also implies the absence of steady state coherence. We conclude that steady state coherence in a a qubit is incompatible with a quantum map.

We explore the industrial and scientific applicability of the VQE-LSTM framework by integrating meta-learning with GPU accelerated quantum simulation using NVIDIA's CUDA-Q (CUDAQ) platform. This work demonstrates how an LSTM-FC meta-initialization module can extend the practical reach of the Variational Quantum Eigensolver (VQE) in both chemistry and physics domains. In the chemical regime, the framework predicts ground-state energies of molecular Hamiltonians derived from PySCF, achieving near FCI accuracy while maintaining favorable O(N^2) scaling with molecular size. In the physical counterpart, we applied the same model to quantized Simple Harmonic Motion systems (SHM), successfully reproducing its ground and excited states through VQE and Variational Quantum Deflation (VQD) methods. Benchmark results on NVIDIA GPUs reveal significant speedups over CPU-based implementations, validating CUDAQ's capability to handle large-scale variational workloads efficiently. Overall, this study establishes VQE-LSTM as a viable and scalable approach for GPU accelerated quantum simulation, bridging quantum chemistry and condensed-matter physics through a unified, meta-learned initialization strategy.

Energy consumption is becoming a serious bottleneck for integrating quantum technologies within the existing global information infrastructure. In photonic architectures, considerable energy overheads stem from using lasers, whose high coherence was long considered indispensable for quantum state preparation. Here, we demonstrate that natural, incoherent sunlight can successfully produce quantum-entangled states via spontaneous parametric down-conversion. We detect polarization-entangled photon pairs with a concurrence of $0.905\pm0.053$ and a Bell state fidelity of $0.939\pm0.027$. Importantly, the system violates Bell's inequality with $S=2.5408\pm0.2171$, exceeding the classical threshold of 2, while maintaining generation rates comparable to laser-based setups. These findings pave the way for sustainable quantum applications in resource-limited environments like interplanetary missions.

Entanglement swapping between photon pairs generated at physically separated nodes over telecommunication fiber infrastructure is an essential step towards the quantum internet, enabling applications such as quantum repeaters, blind quantum computing, distributed quantum computing, and distributed quantum sensing. However, successful networked entanglement swapping relies on generating indistinguishable pairs of photons and preserving them over deployed fibers. This has limited most previous demonstrations to laboratory settings or relied on sophisticated methods to maintain the necessary indistinguishability. Here, we demonstrate a scalable entanglement swapping experiment using naturally indistinguishable entanglement sources based on warm atomic vapor cells. Without sharing lasers or optical frequency references between nodes, nor the need for pulsing the sources, we achieve a swapping rate of nearly 500 pairs/s while maintaining the CHSH parameter above 2. Additionally, we demonstrate the scalability of our method by maintaining the quality of the entanglement swapping on 17.6-km of deployed fibers in NYC, relying on commercially available SPADs at the spoke nodes, SNSPDs at the hub and standard time-synchronization techniques. Our work paves the way for the practical deployment of large-scale hub-and-spoke quantum networks within cities and data centers.

We study correlation properties of polaritons at zero temperature in a multiconnected Jaynes–Cummings (MCJC) lattice on a superconducting circuit quantum electrodynamics platform with engineered cross-Kerr nonlinearity that mimics attractive nearest-neighbour interaction. A multi-connected Jaynes–Cummings lattice is a one-dimensional lattice constructed from alternating qubits and resonators with different left and right couplings. The nearest-neighbour interaction or cross-Kerr coupling is implemented dispersively through ladder-type qutrits between each nearest neighboring pair of resonator modes. Projecting onto the lower-polaritonic manifold, we derive an extended two-mode (bipartite) Bose–Hubbard-like model featuring on-site and attractive nearest-neighbor interactions. Employing a continuum bosonization approach, we express the Hamiltonian in terms of symmetric ($+$) and antisymmetric ($-$) collective modes. In the regime where the ($-$) sector acquires a finite gap, one can reduce the system to an effective single-component Luttinger liquid model for the $+$ sector. The cross-Kerr term reduces the compressibility of the ($+$) mode, thereby enhancing the corresponding Luttinger parameter $K_{+}$, resulting in the slower algebraic decay of single-particle correlations, $G(x)\propto|x|^{-1/(4K_{+})}$.

Nonlinear phase gates are essential to achieve the universality of continuous-variable quantum processing and its applications. We present a deterministic protocol for generating nonlinear phase gates in trapped ion systems using simultaneous two-tone sideband drives beyond the Lamb-Dicke regime. Our approach harnesses higher-order interaction terms typically neglected or suppressed to construct nonlinear phase gates. This methodology enables high-fidelity gate engineering with a near three-fold reduction in control pulses compared to state-of-the-art theoretical proposals.

Electron diffraction from nanogratings provides a platform for free-electron interferometry, yet controlled manipulation of electron spin in such geometries remains largely unexplored. In particular, the role of the self-generated magnetic field arising from electron motion and the feasibility of coherent spin control without disrupting diffraction coherence have not been quantitatively investigated. In this article, a self-consistent Maxwell-Pauli framework is developed to study spin-resolved electron diffraction from nanogratings in the presence of magnetic fields. The model incorporates geometric confinement, image-charge interactions, self-generated magnetostatic fields, and externally applied magnetic fields. Numerical simulations show that the intrinsic magnetic self-field produced by the electron probability current is several orders of magnitude too weak to induce measurable spin mixing, demonstrating that nanogratings act as spin-conserving beam splitters under field-free conditions. When a uniform magnetic field is applied upstream of the nanograting, coherent Larmor precession enables controlled spin rotation without modifying the diffraction geometry or degrading coherence. The magnetic field required for a $\pi$ spin rotation scales inversely with the interaction length and electron de Broglie wavelength $\lambda_{dB}$. Furthermore, a downstream nonuniform magnetic field applied after the nanograting imparts a spatially varying Zeeman phase, producing opposite transverse momentum shifts for the two spin components. The spin-dependent transverse dynamics is analyzed using Husimi Q-function phase-space maps, which visualize spin-dependent population redistribution and momentum separation. The proposed approach enables tunable spatial separation of spin-resolved free electron beams and establishes an all-magnetic route for coherent spin rotation, control, and interferometry.

In systems with fast periodic driving, there are special subsets of (resonant) states, which behavior can be described with effective, time-independent Hamiltonian in a rotating reference frame. Here, we show that experimentally feasible system of ultracold bosonic atoms on a ring with rapidly oscillating scattering length can be used to simulate time-independent two-component atomic mixture with exotic, long-range interactions.

The recent demonstrations of cascaded PDC (CPDC) and the hopeful prospects of realizing third-order PDC (TOPDC) for the generation of three-photon entangled states are paving the way for experimental studies on genuine three-photon interference. In this article, we formulate three-photon interference in terms of “each three-photon interfering only with itself.” We show that although a generalized two-alternative three-photon interference setup based on CPDC or TOPDC involves eight different length parameters, the interference can be fully characterized in terms of only three independent parameters. The first parameter is the three-photon path-length difference, which has a direct analog in the one-photon and two-photon cases, and the other two parameters quantify the path-asymmetry length. Unlike two-photon interference, which requires only one parameter to quantify path-asymmetry, two independent parameters are needed in three-photon interference. This results in a broader class of nonclassical three-photon effects, including three-photon HOM-type effects. Our work provides the theoretical basis for existing and future three-photon interference experiments exploring the rich and complex quantum correlations associated with three-particle entanglement and potentially enabling the development of novel protocols for harnessing those correlations.

We introduce a systematic framework for three-qubit entanglement classification using a cascaded architecture of Support Vector Machine (SVM) classifiers. Leveraging the well defined three-qubit structure with the four nested entanglement classes (S, B, W, and GHZ), we construct three distinct witness models ($\mathcal{M}_{B}$, $\mathcal{M}_{W}$, and $\mathcal{M}_{GHZ}$) that sequentially discriminate between these classes. The proposed Cascaded model achieves an overall classification accuracy of $95\%$ on a comprehensive dataset of mixed states. The framework's robustness and generalization capabilities are confirmed through rigorous testing against out-of-distribution (OOD) entangled states and various quantum noise channels, where the model maintains high performance. A key contribution of this research is an optimization protocol based on systematic feature importance analysis. This approach yields a tunable framework that significantly reduces the number of required features, while maintaining reliable model accuracy.

We present new distributed quantum algorithms for fundamental distributed computing problems, namely, leader election, broadcast, Minimum Spanning Tree (MST), and Breadth-First Search (BFS) tree, in arbitrary networks. These algorithms are (essentially) optimal with respect to their communication (message) complexity in the quantum routing model introduced in [PODC 2025]. The message complexity of our algorithms is $\tilde{O}(n)$ for leader election, broadcast, and MST, and $\tilde{O}(\sqrt{mn})$ for BFS ($n$ and $m$ are the number of nodes and edges of the network, respectively). These message bounds are nearly tight in the quantum routing model since we show almost matching corresponding quantum message lower bounds. Our results significantly improve on the prior work of [PODC 2025], who presented distributed quantum algorithms under the same model that had a message complexity of $\tilde{O}(\sqrt{mn})$ for leader election. Our algorithms demonstrate the significant communication advantage that quantum routing has over classical in distributed computing, since $\Omega(m)$ is a well-established classical message lower bound for leader election, broadcast, MST, and BFS that applies even to randomized Monte-Carlo algorithms [JACM 2015]. Thus, our quantum algorithms can, in general, give a quadratic advantage in the communication cost for these fundamental problems. A main technical tool we use to design our distributed algorithms is quantum walks based on electric networks. We posit a framework for using quantum walks in the distributed setting to design communication-efficient distributed quantum algorithms. Our framework can be used as a black box to significantly reduce communication costs and may be of independent interest. Additionally, our lower-bound technique for establishing distributed quantum message lower bounds can also be applied to other problems.

We digitally simulate quantum many-body dynamics in emergent curved backgrounds using 80 superconducting qubits on IBM Heron processors. By engineering spatially varying couplings in the spin-$\frac12$ XXZ chain, consistent with the low-energy description of the model in terms of an inhomogeneous Tomonaga-Luttinger liquid, we realize excitations that follow geodesics of an effective metric inherited from the underlying spatial deformation. Following quenches from Néel and few-spin-flip states, we observe curved light-cone propagation, horizon-induced freezing in the local magnetization, and position-dependent oscillation frequencies set by the engineered spatial deformation. Despite strong spatial inhomogeneity, unequal-time correlators reveal ballistic quasiparticle propagation in the spin chain. These results establish large-scale digital quantum processors as a flexible platform for detailed and controlled exploration of many-body dynamics in tunable and synthetic curved spacetimes.

Quantum correlations, crucial for the advantage and advancement of quantum science and technology, arise from the impossibility of expressing a quantum state as a tensor product over a given set of parties. In this work, a generalized notion of correlations via arbitrary products is formulated. Remarkably, as a universal property, the connection between such general products and tensor products is established, allowing one to relate generic non-product states to the common notion of entangled states. We construct the set of free operations for general types of products by extending the local-operation-and-classical-communication paradigm, familiar from standard entanglement theory, thereby establishing a resource theory of correlations for general products. A generalization is provided beyond two factors that can be universally related to multipartite entanglement. Applications that highlight the usefulness of the approach are discussed, such as the factorization of fermionic states, the non-local factorization of multi-photon states into single-photon states, and the interesting possibility of understanding prime numbers as a form of single-party entanglement.

Quantum architectures based on neutral atoms have gained significant attention in recent years as specialized computational machines due to their ability to directly encode the independent set constraint on graphs, exploiting the Rydberg blockade mechanism. In this work, we address the Drone Delivery Packing Problem via a hybrid quantum-classical framework leveraging a neutral-atom quantum processing unit (QPU). We reformulate the optimization task as a graph-partitioning problem based on the independent sets (ISs) of a scheduling graph that encodes delivery incompatibilities. Each partition corresponds to deliveries assigned to a single drone, with the objective of minimizing the total number of partitions. While the ISs represent time-feasible schedules, battery-duration constraints are enforced through a classical post-processing routine. This methodology enables the recovery of optimal delivery schedules, provided a sufficient number of samples is collected from the QPU to resolve the solution space. We benchmark the hybrid workflow through numerical emulations and demonstrate its effectiveness on Pasqal's Fresnel QPU, reporting hardware experiments with configurations of up to 100 atoms.

We analyze numerically the performance of Quantum Reservoir Computing (QRC) for statistical and financial problems. We use a reservoir composed of two superconducting islands coupled via their charge degrees of freedom. The key non-linear elements that provide the reservoir with rich and complex dynamics are the Josephson junctions that connect each island to the ground. We show that QRC implemented in this circuit can accurately classify complex probability distributions, including those with heavy tails, and identify regimes in correlated time series, such as periods of high volatility generated by standard econometric models. We find QRC to outperform some of the best classical methods when the amount of information is limited. This demonstrates its potential to be a noise-resilient quantum learning approach capable of tackling real-world problems within currently available superconducting platforms. We further discuss how to improve our QRC algorithm in real superconducting hardware to benefit from a much larger Hilbert space.

Quantum batteries, miniaturized devices able to store and release energy on demand, are promising both because their intrinsic energy and time scales can match those of other quantum technologies and due to the intriguing possibility of achieving super-extensive charging power. While this enhanced scaling is known to appear in several settings, it is generally believed to be forbidden in Jordan-Wigner integrable spin chains charged via quantum-quench protocols. Here, we show that an extended cluster-Ising model, despite belonging to the above category, exhibits super-extensive charging power over wide ranges of system sizes, reaching up to a thousand spins, in proper parameter regimes. This remarkable anomalous scaling is due to a corresponding super-extensive growth of the stored energy, implying that it is limited to a large but finite size of the system and cannot persist in the thermodynamic limit. This phenomenon appears robust against finite-temperature effects.

We study the task of quantum state exclusion, focusing on antidistinguishability and its generalization to $x$-antidistinguishability, under global measurements and local operations with classical communication (LOCC). We also introduce weak and strong notions of antidistinguiahbaility ($x$-antidistinguishability) depending on whether all states or all $x$-tuples are exhaustively eliminated. Our results reveal striking differences between state exclusion and the more familiar task of state discrimination. In particular, we show that LOCC antidistinguishability of multipartite product states is symmetric with respect to the initiating party but this symmetry breaks down for higher-order $x$-antidistinguishability. Most notably, we establish a manifestation of nonlocality without entanglement in the context of state exclusion: we prove that three bipartite product states can be globally antidistinguishable while failing to be LOCC antidistinguishable, demonstrating that three is the minimal number of states required for this phenomenon. We further extend this separation to $2$-antidistinguishability and present example exhibiting the same type of nonlocality. At last, we provide an antidistinguishable tripartite product states that are not LOCC antidistinguishable across any bipartition, which ensures the phenomenon of genuine nonlocality without entanglement in this framework.

Deep generative modeling to stochastically design small molecules is an emerging technology for accelerating drug discovery and development. However, one major issue in molecular generative models is their lower frequency of drug-like compounds. To resolve this problem, we developed a novel framework for optimization of deep generative models integrated with a D-Wave quantum annealing computer, where our Neural Hash Function (NHF) presented herein is used both as the regularization and binarization schemes simultaneously, of which the latter is for transformation between continuous and discrete signals of the classical and quantum neural networks, respectively, in the error evaluation (i.e., objective) function. The compounds generated via the quantum-annealing generative models exhibited higher quality in both validity and drug-likeness than those generated via the fully-classical models, and was further indicated to exceed even the training data in terms of drug-likeness features, without any restraints and conditions to deliberately induce such an optimization. These results indicated an advantage of quantum annealing to aim at a stochastic generator integrated with our novel neural network architectures, for the extended performance of feature space sampling and extraction of characteristic features in drug design.

In this manuscript, we study the Schrödinger cat state transfer in a quantum optical version of Newton's cradle in non-Markovian environment. Based on a non-Markovian master equation, we show that the cat state can be transferred purely through the memory effect of the non-Markovian common environment, even without any direct couplings between neighbor cavities. The mechanism of the environment induced cat state transfer is analyzed both analytically and numerically to demonstrate that the transfer is a unique phenomenon in non-Markovian regime. From this example, the non-Markovian environment is shown to be qualitatively different from the Markovian environment reflected by the finite versus zero residue coherence. Besides, we also show the influence of environmental parameters are crucial for the transfer. We hope the cat state transfer studied in this work may shed more light on the fundamental difference between non-Markovian and Markovian environments.

We investigate how quasiperiodic spatial structure enables protocol-induced acceleration in open quantum systems by analyzing the Pontus-Mpemba effect in one-dimensional chains subject to Markovian dephasing. The dynamics are governed by a Lindblad superoperator that drives all initial states toward a maximally mixed infinite-temperature steady state, isolating dynamical mechanisms from static equilibrium properties. Considering two representative quasiperiodic models, namely a tight-binding chain with a mosaic potential and its extension with power-law long-range hopping, we show that a properly engineered two-step protocol, in which the system is first steered to a finite temperature intermediate state, yields a strictly shorter overall relaxation time than direct evolution from the same initial configuration. This protocol-induced acceleration persists for both initially localized and extended eigenstates and remains robust in the presence of long-range hopping. A Liouvillian spectral analysis reveals that the mechanism originates from a redistribution of spectral weight that suppresses overlap with the slowest decay modes, rather than from any modification of the decay spectrum itself. Our results establish quasiperiodic chains as a controlled setting for engineering relaxation pathways through Liouvillian spectral structure.

In traditional research, chaos is frequently accompanied by non-linearity, which typically stems from non-linear interactions or external driving forces. However, in this paper, we present the chaotic behavior that is completely attributed to the non-linear back-reaction of non-Markovian environment. To be specific, we derive the dynamical equations of an optomechanical system and demonstrate that the non-linearity (cause of chaos) in the equations arises entirely from the time-domain convolutions (TDCs) induced by non-Markovian corrections. Under Markovian conditions, these TDCs are reduced into constants, thereby losing the nonlinearity and ultimately leading to the disappearance of chaos. Furthermore, we also observe chaos generation in the absence of optomechanical couplings, which further confirms that the non-Markovian effect is the sole inducement of chaos and the environmental parameters play important roles in the generation of chaos. We hope these results may open a new direction to investigate chaotic dynamics purely caused by non-Markovian environments.

We propose a stochastic Schrödinger equation (SSE) approach to investigate the dynamics of giant atoms coupled to a waveguide, addressing two critical gaps in existing research, namely insufficient exploration on continuous coupling and multiple excitations. A key finding is that continuous coupling, unlike discrete coupling at finite points, breaks the constant phase difference condition, thereby weakening the interference effects in giant atom-waveguide systems. In addition, a key technical advantage of the SSE approach is that auto- and cross-correlation functions can directly reflect the complex photon emission/absorption processes and time-delay effects in giant atom-waveguide systems. Moreover, the SSE approach also naturally handles multiple excitations, without increasing equation complexity as the number of excitations grows. This feature enables the investigation of multi-excitation initial states of the waveguide, such as thermal and squeezed initial states. Overall, our approach provides a powerful tool for studying the dynamics of giant atoms coupled to waveguide, particularly for continuous coupling and multi-excitation systems.

Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.

We investigate the capability of multiple-braiding in $SU(2)_k$ anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the $q$-deformed representation theory of $SU(2)$. Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity $m$ causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of $SU(2)_k$ anyon models with $k = 3, 5, 6, 7$, and $m \in [2, 9]$, shows that the values of $m$ at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those $m$ that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class $[CNOT]$. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities $m$ support universal quantum computation in $SU(2)_k$ anyon models.

Fault-tolerant quantum computation requires substantial resources for non-Clifford gates. Their count also enters exponentially into the runtime of several classical simulation methods. We present algebraic methods for minimizing the non-Clifford CCZ (Toffoli) count in {CNOT,CCZ} phase circuits using tensor decomposition over $\mathbb{F}_2$. On standard cubic benchmarks, these methods match or improve prior automated Toffoli counts, with most instances optimized in under a minute on a single CPU. With additional computation, the decompositions provide effective initializations for existing T-count methods, improving T counts on several circuits. We also present algebraic methods for CNOT-count optimization at fixed Toffoli count. This gives competitive ancilla-free finite-field multiplication circuits and reduces the CNOT count of XOR AES S-box circuits.

We present an experimental characterization of coexistence-induced interference in QKD transmission arising from SpRS and FWM, validating a comprehensive semi-analytical model for accurate noise estimation. Experimental results show good agreement with theoretical predictions.

We investigate whether the spinor field can be differential-algebraically eliminated from the Maxwell–Dirac equations in a particular gauge. To this end, we construct a generic truncated power-series solution and linearize the prolonged system of the Maxwell–Dirac equations about this solution. We then analyze the ranks of the coefficient matrices associated with the linearized system. Our results indicate that, generically, the spinor components are uniquely determined by the electromagnetic field and its derivatives. Furthermore, the fourth-order time derivatives of the components of the electromagnetic four-potential are uniquely determined by derivatives of the lower order with respect to time. These findings strongly suggest that the spinor field can be differential-algebraically eliminated, and the resulting equations describe independent evolution of the electromagnetic field, i.e., a Cauchy problem can be formulated in terms of the electromagnetic variables alone.

We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.